Non-linear structure in the Universe Cosmology on the Beach

Transcription

1 Non-linear structure in the Universe Cosmology on the Beach Puerto Vallarta January, 2011 Martin White UC Berkeley/LBNL (

2 Strong non-linearity Martin White UCB/LBNL

3 Limited options Beyond a certain scale, linear perturbation theory breaks down Definition of non-linear scale? At this point we have few options: Analytical models of non-linear growth. Zel dovich approximation. Spherical top-hat collapse. Perturbation theory. Realm of validity? Convergence criterion? Good for small corrections to almost linear problems. Direct simulation. Numerical convergence. What models to run? Missing physics.

4 Zel dovich approximation Assume particles move in a straight line with their linear perturbation theory velocity. Defines a mapping from initial (Lagrangian) position, q, to final (Eulerian) position, x: x=q+ψ with Ψ(q,t)=D(t)Ψ(q) and Ψ i =dφ/dq i Ψ k = -ik/k 2 δ k If the initial field is uniform, the final density is the Jacobian of this mapping. ρ~[(1-dα)(1-dβ)(1-dγ)] -1 α,β,γ e-values of d 2 Φ/dq i dq j Collapse takes place first along largest e- value (pancake/sheet), then middle (filament) then final (halo).

5 The cosmic web The Zel dovich approximation, plus the statistics of Gaussian fields, qualitatively describes large-scale structure.

6 Numerical simulations Our ability to simulate structure formation has increased tremendously in the last decade. Direct simulation of the N-body problem Begin at early times, but during matter domination, by displacing particles from an initial grid using 1LPT or 2LPT. Monte-Carlo integration of the Vlasov equation using super-particles which move along the characteristics. Soften the forces to avoid particle-particle scattering or integrating unphysical, tight, orbiting particles. Want to approach the fluid limit with very large N. Pure N-body codes scale almost perfectly. Our understanding of -- or at least our ability to describe -- galaxy formation has also increased dramatically. Most cosmology probes observe galaxies. The fundamental unit of structure theoretically is the dark matter halo. Galaxies live in dark matter halos in ways we increasingly understand.

7 Numerical convergence Numerous tests of numerical convergence can be found in: Heitmann et al. (2010; ApJ, 715, 104) Heitmann et al. (2010; ApJ, 705, 156) Need to worry about Starting redshift and method. Force accuracy and softening. Time stepping. Box size. Number of particles. Method of computing statistic from particles. How to choose which cosmologies to run.

8 Accuracy - currently demonstrated Updated from Heitmann et al. (2007) Only a subsample of the codes are shown here. All codes started from the same ICs and analyzed with the same P(k) codes.

9 Extra physics As we go to smaller scales, we must go beyond the pure N- body problem and include additional physics. Hydrodynamics solvers well developed. Gas cools dramatically in deep potential wells, reaching high densities in a clumpy, multiphase, turbulent, magnetized ISM where it can form stars, which give off winds and radiation and go supernova injecting momentum and energy into the surrounds and have active galactic nuclei which can impart energy to their enviroments, There is little scale separation between including gas physics and including star formation, feedback, etc. so results typically depend on sub-grid models.

10 An example One possibility, from Jing et al. (2006), for the effects of baryons (red) and baryons including starformation and feedback (green) on the total matter (solid), dark matter (dotted) and gas (dashed).

11 Characteristics of LSS Large-scale structure forms a beaded filamentary web of dark matter halos. Number of halos vs. mass (etc.). Spatial distribution of halos (vs.?). Properties of DM halos. Beyond DM.

12 Halo abundance Almost all of the mass resides in (approximately) virialized halos. Space density of halos depends primarily (exclusively?) on mass. There are a large number of low mass halos and few high mass halos. Very roughly dn ~ m -2 e -m As time proceeds the characteristic mass scale increases. The mass function is almost cosmology independent (in scaled units). This universality is not fully understood. Mass functions are used in many applications in cosmology.

13 Mass function Bhattacharya++10 Note the dynamic range in this figure!

14 Halo abundance: scaled units dn dm = f(σ) ρ M d ln σ 1 dm Tinker++08

15 Other fitting forms (A detailed study of universality and numerical issues can be found in Bhattacharya++10 from which this table is taken ) MASS FUNCTION FITTING FORMULAE DERIVED IN PREVIOUS STUDIES Reference Fitting function f(σ) MassRange Redshiftrange [ ] [ ( ) ] 0.3 2(0.75) Sheth & Tormen (2002) f ST (σ)= π exp! 0.75δ2 c δ 2σ 1 + c 2 σ Unspecified Unspecified Jenkins et al. (2001) 0.315exp [! lnσ! ]!1.2 lnσ! z= 0-5 σ δ 2 c Warren et al. (2006) ( σ! ) exp [ ]! σ 2 [ ( 2(0.707) Reed et al. (2007) π 1 + Manera et al. (2010) Crocce et al. (2010) σ δ 2 c ) G1 (σ) + 0.4G 2 (σ) [ ] δ c σ exp! 0.764δ2 c σ! 2 (n ef f +3) 2 (δ c /σ) 0.6 [ 2a f ST (σ)= π exp! aδ2 c 2σ ][1 + 2 ( A(z) [ σ!a(z) + b(z) ] exp [! c(z) σ 2 ] σ 2 aδ 2 c ] (10 10! )h!1 M z=0!0.5 lnσ!1 1.2 z=0-30 ) p ] δc σ ( ! )h!1 M z=0-0.5 (10 10! )h!1 M z=0-1 f(σ) = M ρ dn d ln σ 1, 0 d ln σ f(σ) =1

16 Excursion set theory vs. peaks Excursion set formalism The most popular theory. The fraction of mass in halos more massive than M is related to the fraction of volume in which the smoothed initial density field is above some threshold, δ c. Mass function related to random walk. Press-Schechter 1974; Bond, Cole, Efstathiou & Kaiser Spherical collapse vs. elliptical collapse approx. Mo & White, Sheth & Tormen, Zhang & Lam, How to deal with non-locality of halo collapse. Statistics of (Gaussian) peaks plus a model for halo collapse (spherical or ellipsoidal). Bardeen, Bond, Kaiser & Szalay 1986 Based on Rice (1944; 1945) who studied 1D Gaussian fields as models of noise in communications devices. Bond & Myers Dalal, Lithwick & White 201X.

17 Excursion set theory vs. peaks Allow computation of mass function from statistics of initial field. Choose a filter shape, and compute integrals of linear theory power spectrum and plug in formulae. Not all methods self-consistent. Reasonable success for mass function often improved by adjusting formulae to fit N-body simulations. Less success for conditional mass function, merger rates etc. Beware when extrapolating!

18 Halo bias The clustering of the rare, massive dark matter halos is enhanced relative to the general mass distribution Kaiser 1984; Efstathiou++88; Cole & Kaiser 1989; Bond++91; Mo & White 1996; Sheth & Tormen 1999; ; Tinker++10;... The clustering of rare halos thought to host quasars (here and M sun /h) at z=3-4 is two orders of magnitude stronger than the clustering of the DM!

19 Halo bias This enhanced clustering is known as bias. Bias depends on scale [b(r)], but at very large scales it becomes scaleindependent [b]. Bias, b, depends primarily on halo mass or rarity. In simplest models b=1+(ν 2-1)/δ c, where ν=δ c /σ(m). For more accuracy, use N-body-calibrated fitting function. Behavior at extremes can depart from fitting functions! Numerical simulations now large enough to test for the dependence on halo formation history and other properties. Dependencies on formation redshift, internal structure, and spin. Gao++05; Wechsler++06; Harker++06; Bett++07; Wetzel++07; Jing++07; Gao&White07; Angulo++08

20 Halo bias in simulations Tinker++10 Halo bias increases with increasing halo mass at fixed redshift, or with increasing redshift at fixed mass.

21 Assembly bias Gao & White (2007) Solid (dashed) lines show halos in lower (upper) 20% of halos split on property labeled.

22 Assembly bias Assembly bias is quite difficult to explain in the standard excursion set formulation. Mass function is fraction of random walks reaching an absorbing barrier by mass M. Bias is dependence of mass function on large-scale density (early part of the walk). Assembly bias very hard to explain in this picture. Gao++05, Mo++05, Sandvik++07, Desjacques08, Simulations did not initially shed light on explanation for assembly bias. Now understand that assembly bias is a simple consequence of non-linear collapse from Gaussian initial conditions. Dalal++08.

23 Assembly bias: high mass. Later forming, high mass halos are more clustered than typical halos of the same mass. Also dependence on concentration. Massive halos collapse almost spherically from rare peaks in ICs. Collapse reasonably explained by STHC. For Gaussian field, bias depends on curvature, s=d<δ>/dlnm, of peak (as well as height). Peak curvature is environment : δ b =δ pk + s dlnm + Peaks with smaller s have larger background densities. b 1 1 σ ν νx x 1 νx 2, ν δ σ δ ; x s σ s (Cross-correlation coefficient)

24 Bias: high mass b E 9.0 Dalal++08 Dependence of halo bias 8.5 on peak curvature from 8.0 simulations (points) compared to the 7.5 prediction from Gaussian 7.0 peaks theory (line) for a 6.5 power-law model. Assembly history related 6.0 to run of δ with M 5.5 accretion rate related to peak curvature! 5.0!0.5!0.4!0.3!0.2!0.1 d(log!)/d(log M) ~ -[d(logm)/d(log a)] -1

25 Assembly bias: low mass Oldest, most concentrated, low mass halos are more than twice as clustered as the youngest halos of the same mass. Youngest ~80% of halos have b~1-δ c -1 ~0.4 (as expected). Oldest 20% of low mass halos act like test particles (b >1) Most of these are associated with nearby, high-mass halos. Early formers who s growth is stunted by hot environments of massive neighbors.

26 DM halos Generally triaxial spheroids. More elongated at Smaller radii. Larger redshifts. Higher mass. Approximately in virial equilibrium. Aligned with the filamentary, cosmic web which feeds halo growth. Average mass accretion exponential. In EPS formalism dm=-f(m)m dδ c, with f(m)~constant. Spin parameter, λ, grows significantly in major mergers, slowly declines in accretion.

27 Dynamical state (White 2002) Roughly isothermal, roughly virialized, self-bound objects.

28 DM halos are aspherical and have significant substructure Region above a density 10 2 times the background density. Color: logdensity.

29 Spherical NFW profile ρ(x = r/r s ) x 1 (1 + x) 2 Mean interior density Tinker++08

30 A 1-parameter family Find c=r vir /r s is a function of M. More massive halos less concentrated. c, like M, depends on definitions! c~m Large, log-normal scatter in c. The inner, r -1, part of the halo forms early and r s stays ~constant. Subsequent accretion kept away by angular momentum barrier. Concentration, c=r vir /r s ~ (1+z) -1.

31 Other forms A generalized NFW makes -1 and -2 variable. Einasto profile: ρ exp d n r r e 1/n 1 Note no cusp! Important new insights in Lithwick & Dalal (2010). Building on earlier work by Fillmore & Goldreich and Bertschinger. The NFW profile is transitional. r -3 slope comes from continued accretion of material. This stops in DE-domination. Busha, Evrard & Adams (2007)., n 5 10 Exponential truncation of NFW profile at large radius.

32 Subhalos A generic prediction of hierarchical theories, such as CDM, is that the virialized regions of DM halos contain subhalos. Self-gravitating, bound clumps of mass. Subhalos account for O(10%) of halo mass. Luminous galaxies form via the cooling and condensation of gas in subhalos.

33 Subhalos Density profiles of subhalos similar to that of halos, but they can be truncated. Subhalos track DM closely in terms of density and velocity. Trends of central concentration and velocity bias with ratio of subhalo to host halo mass. Depends on how subhalos are selected. Beyond a certain point, the number of subhalos above a given mass grows linearly with host halo mass. Length of plateau set by dynamical friction and mean density of collapsed structures. Subhalo mass function and halo mass function are scaled versions of each other. dn sat dm sat Mhost M sat 2, M sat M host

34 Thank you! I would like to thank The participants. The other lecturers. The organizers. The staff. for making this a pleasant, informative and productive meeting.

35 The End